Export video from Game Engine scene
There is no checkbox called "Dump framebuffer to disk", so you will have to use a script for this.

The core idea from that old forum thread is good, call bge.render.makeScreenshot() every frame with different filenames.

In the Text Editor, make a script file with a name like "screenshooter.py". That script will handle a frame counter, will make a filename based on that frame counter, and will give that filename to makeScreenshot().

from bge import render

counter = 0

def shoot():
    global counter

    filename = "bge_rendered/frame_" + str(counter).zfill(8) + ".png"
    print("Saving to " + filename)
    render.makeScreenshot(filename)

    counter += 1


In the Logic Editor make:


an Always sensor, with "Activate TRUE level triggering".
a Python controller, set to Module execution method, executing "screenshooter.shoot".
the two connected together.


In the render settings, disable "Use Frame Rate" to disable frame-skipping, otherwise there will be a lot of duplicated frames and jumps in the animation.